In the context of differential geometry, what is a function that separates points of a manifold?
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If you have a family $\mathcal F$ of functions from a set $X$ into some other set, we say that it separates the points of $x$ if, for each $x_1,x_2\in X$ with $x_1\neq x_2$, there is some $f\in\mathcal F$ such that $f(x_1)\neq f(x_2)$.
So, if $\mathcal F$ consists of a single function $f$, this is the same thing as asserting that $f$ is injective.
A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $0$ at one point and $1$ at the other.
Let $A$ and $B$ be sets and let $S$ be a set of functions $f:A \to B.$ $S$ is said to separate the points of $A$, if for any $x,y \in A$ with $x \ne y$, there is $f \in S$ such that $f(x) \ne f(y)$.